Theorem natoppfb 49721

Description: A natural transformation is natural between opposite functors, and vice versa. (Contributed by Zhi Wang, 18-Nov-2025.)

Hypotheses

Ref	Expression
natoppf.o	⊢ 𝑂 = (oppCat‘𝐶)
natoppf.p	⊢ 𝑃 = (oppCat‘𝐷)
natoppf.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
natoppf.m	⊢ 𝑀 = (𝑂 Nat 𝑃)
natoppfb.k	⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
natoppfb.l	⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
natoppfb.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
natoppfb.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
natoppfb	⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))

Proof of Theorem natoppfb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		natoppf.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
2		natoppf.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐷)
3		natoppf.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		natoppf.m	. . . 4 ⊢ 𝑀 = (𝑂 Nat 𝑃)
5		natoppfb.k	. . . . 5 ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
6	5	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝐾 = ( oppFunc ‘𝐹))
7		natoppfb.l	. . . . 5 ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
8	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝐿 = ( oppFunc ‘𝐺))
9		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝑥 ∈ (𝐹𝑁𝐺))
10	1, 2, 3, 4, 6, 8, 9	natoppf2 49720	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝑥 ∈ (𝐿𝑀𝐾))
11		eqid 2739	. . . . 5 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂)
12		eqid 2739	. . . . 5 ⊢ (oppCat‘𝑃) = (oppCat‘𝑃)
13		eqid 2739	. . . . 5 ⊢ ((oppCat‘𝑂) Nat (oppCat‘𝑃)) = ((oppCat‘𝑂) Nat (oppCat‘𝑃))
14	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐿 = ( oppFunc ‘𝐺))
15	14	fveq2d 6831	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐿) = ( oppFunc ‘( oppFunc ‘𝐺)))
16	4	natrcl 17911	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐿𝑀𝐾) → (𝐿 ∈ (𝑂 Func 𝑃) ∧ 𝐾 ∈ (𝑂 Func 𝑃)))
17	16	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐿 ∈ (𝑂 Func 𝑃) ∧ 𝐾 ∈ (𝑂 Func 𝑃)))
18	17	simpld 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐿 ∈ (𝑂 Func 𝑃))
19	14, 18	eqeltrrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐺) ∈ (𝑂 Func 𝑃))
20		relfunc 17820	. . . . . . 7 ⊢ Rel (𝑂 Func 𝑃)
21		eqid 2739	. . . . . . 7 ⊢ ( oppFunc ‘𝐺) = ( oppFunc ‘𝐺)
22	19, 20, 21	2oppf 49622	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘( oppFunc ‘𝐺)) = 𝐺)
23	15, 22	eqtr2d 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐺 = ( oppFunc ‘𝐿))
24	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐾 = ( oppFunc ‘𝐹))
25	24	fveq2d 6831	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐾) = ( oppFunc ‘( oppFunc ‘𝐹)))
26	17	simprd 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐾 ∈ (𝑂 Func 𝑃))
27	24, 26	eqeltrrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))
28		eqid 2739	. . . . . . 7 ⊢ ( oppFunc ‘𝐹) = ( oppFunc ‘𝐹)
29	27, 20, 28	2oppf 49622	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘( oppFunc ‘𝐹)) = 𝐹)
30	25, 29	eqtr2d 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐹 = ( oppFunc ‘𝐾))
31		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐿𝑀𝐾))
32	11, 12, 4, 13, 23, 30, 31	natoppf2 49720	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐹((oppCat‘𝑂) Nat (oppCat‘𝑃))𝐺))
33	1	2oppchomf 17681	. . . . . . . 8 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))
34	33	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)))
35	1	2oppccomf 17682	. . . . . . . 8 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))
36	35	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (compf‘𝐶) = (compf‘(oppCat‘𝑂)))
37	2	2oppchomf 17681	. . . . . . . 8 ⊢ (Homf ‘𝐷) = (Homf ‘(oppCat‘𝑃))
38	37	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (Homf ‘𝐷) = (Homf ‘(oppCat‘𝑃)))
39	2	2oppccomf 17682	. . . . . . . 8 ⊢ (compf‘𝐷) = (compf‘(oppCat‘𝑃))
40	39	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (compf‘𝐷) = (compf‘(oppCat‘𝑃)))
41		natoppfb.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
42	41	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐶 ∈ 𝑉)
43		natoppfb.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
44	43	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐷 ∈ 𝑊)
45	1, 2, 42, 44, 27	funcoppc5 49635	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐹 ∈ (𝐶 Func 𝐷))
46	45	func1st2nd 49566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
47	46	funcrcl2 49569	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐶 ∈ Cat)
48	1	oppccat 17679	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
49	11	oppccat 17679	. . . . . . . 8 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat)
50	47, 48, 49	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (oppCat‘𝑂) ∈ Cat)
51	46	funcrcl3 49570	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐷 ∈ Cat)
52	2	oppccat 17679	. . . . . . . 8 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
53	12	oppccat 17679	. . . . . . . 8 ⊢ (𝑃 ∈ Cat → (oppCat‘𝑃) ∈ Cat)
54	51, 52, 53	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (oppCat‘𝑃) ∈ Cat)
55	34, 36, 38, 40, 47, 50, 51, 54	natpropd 17937	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐶 Nat 𝐷) = ((oppCat‘𝑂) Nat (oppCat‘𝑃)))
56	3, 55	eqtrid 2786	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑁 = ((oppCat‘𝑂) Nat (oppCat‘𝑃)))
57	56	oveqd 7373	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐹𝑁𝐺) = (𝐹((oppCat‘𝑂) Nat (oppCat‘𝑃))𝐺))
58	32, 57	eleqtrrd 2842	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐹𝑁𝐺))
59	10, 58	impbida 806	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐹𝑁𝐺) ↔ 𝑥 ∈ (𝐿𝑀𝐾)))
60	59	eqrdv 2737	1 ⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ‘cfv 6485  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Catccat 17621  Hom_f chomf 17623  comp_fccomf 17624  oppCatcoppc 17668   Func cfunc 17812   Nat cnat 17902   oppFunc coppf 49612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-homf 17627  df-comf 17628  df-oppc 17669  df-func 17816  df-nat 17904  df-oppf 49613

This theorem is referenced by:  fucoppclem  49897  lmddu  50157